Finite Group Databasebeta

← #27370090

Subgroups — #27370090

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285840C211yesmaximal, characteristic, complemented
2C2×#16810828C211yesmaximal, characteristic, complemented
3#20283750C211yesmaximal, characteristic, complemented
4#20274274C212yesmaximal, complemented
5#20274216C212yesmaximal, complemented
6#20274150C212yesmaximal, complemented
7#20274092C212yesmaximal, complemented
8#20273501C211yesmaximal, characteristic, complemented
9#20273433C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3109C211yesmaximal, characteristic
12C22×#11719923—21no—
13C2×#13133217C2211yescharacteristic
14#16811019C2212yescomplemented
15#16810969C2212yescomplemented
16#16810939—21no—
17#16810939—21no—
18#16810913—22no—
19#16810828—21no—
20#16810828—21no—
21#16810828C2212yes—
22#16810828C2212yes—
23#16810701C2212yes—
24#16810695—22no—
25D4×#88322C2212yes—
26#16810680C2211yescharacteristic
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133140—21no—
30#16810587—21no—
31C2×C22⋊#88322—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810372C2212yescomplemented
35#16810370C2212yescomplemented
36#16810370C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#532C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#481—21no—
41C53×#476C2211yescharacteristic
42C53×#476C2211yescharacteristic
43C53×#459C2211yescharacteristic
44C53×#459C2211yescharacteristic
45C53×#447—21no—
46C53×#442—21no—
47C53×#442—21no—
48C53×#432C2211yescharacteristic
49C22×Dic106—21no—
50C2×#11719923—22no—
51C2×#11719923—22no—
52C2×#11719923—44no—
53C2×#11719917—22no—
54C22×#88322—22no—
55C22×#88322—21no—
56#13133217—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217C2312yes—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62#13133148—22no—
63#13133148—22no—
64C4⋊#88322—21no—
65C4⋊#88322D412yes—
66C4⋊#88322C2312yes—
67C4⋊#88322C2312yes—
68#13133140—22no—
69#13133140—22no—
70#13133138—22no—
71#13133138—22no—
72C22⋊#88322—22no—
73C22⋊#88322—22no—
74C22⋊#88322—21no—
75C22⋊#88322D411yescharacteristic
76C22⋊#88322D411yescharacteristic
77#13133128—21no—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C22×D4×C53D411yescharacteristic
81C23×C4×C53—21no—
82C2×D8×C53—21no—
83C2×D8×C53—21no—
84C4⋊Q8×C53C2311yescharacteristic
85#127×C53C2311yescharacteristic
86#124×C53—21no—
87#121×C53—21no—
88#121×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53D411yescharacteristic
94C4×D4×C53D411yescharacteristic
95C4×D4×C53C2311yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C22⋊C4×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#106×C53—21no—
101#106×C53—21no—
102C4⋊C8×C53C2311yescharacteristic
103C4⋊C8×C53C2311yescharacteristic
104#102×C53—21no—
105#102×C53—21no—
106#102×C53—21no—
107#102×C53—21no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C24×C53—21no—
111#41×C53—82no—
112C2×Q8×C53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C2×D4×C53—41no—
120C2×D4×C53C2×D411yescharacteristic
121C2×D4×C53C2×D411yescharacteristic
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—41no—
128C22×C4×C53—82no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C22×Dic53—41no—
133C2×Dic106—22no—
134C2×Dic106—42no—
135C2×Dic106—42no—
136C2×Dic106—41no—
137D8×C53—42no—
138D8×C53—42no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53C2×D411yescharacteristic
145C4⋊C4×C53C2×D411yescharacteristic
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53#4111yescharacteristic
155C42×C53C2411yescharacteristic
156#11719923—44no—
157#11719923—44no—
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53—41no—
161C4⋊Dic53—41no—
162#11719917—82no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—42no—
166C2×#88322—82no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—42no—
172C23×C53—42no—
173C23×C53—41no—
174C23×C53—81no—
175C23×C53#12111yescharacteristic
176Q8×C53—81no—
177D4×C53—21no—
178D4×C53—21no—
179D4×C53—21no—
180D4×C53—42no—
181D4×C53—41no—
182D4×C53—41no—
183D4×C53—42no—
184D4×C53—81no—
185D4×C53—82no—
186D4×C53AGL(1,ℤ/8)12yes—
187C2×C4×C53—22no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—41no—
196C2×C4×C53—81no—
197C2×C4×C53—81no—
198C2×C4×C53—82no—
199C2×C4×C53—81no—
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C2×Dic53—42no—
207C2×Dic53—82no—
208Dic106—42no—
209Dic106—42no—
210Dic106—82no—
211C8×C53—41no—
212C8×C53—41no—
213#88322—22no—
214#88322—22no—
215#88322—42no—
216#88322—42no—
217#19462—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—22no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53—84no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53—81no—
230C22×C53#47112yes—
231C22×C53C2×#12111yescharacteristic
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—41no—
237C4×C53—41no—
238C4×C53—81no—
239C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C53#57511yescharacteristic
241C4×C53#57712yes—
242Dic53—82no—
243#3227—531no—
244#3109D5311yescharacteristic
245C2×#461—531no—
246#2974—531no—
247#1701—532no—
248#1691—532no—
249#1681—532no—
250#1671—532no—
251#1573—531no—
252#1556—531no—
253#1551—531no—
254C2×C53—22no—
255C2×C53—42no—
256C2×C53—81no—
257C2×C53—161no—
258C2×C53#304211yescharacteristic
259C2×C53#305511yescharacteristic
260C2×C53C2×#47111yescharacteristic
261C22×SD16—1061no—
262#532C2×D5311yescharacteristic
263C2×C4⋊Q8—531no—
264C2×C4×D4C2×D5311yescharacteristic
265#498—532no—
266#486—532no—
267#481—21no—
268#481—1061no—
269#481—1061no—
270#476—1062no—
271#476C2×D5311yescharacteristic
272#476C2×D5311yescharacteristic
273#461—532no—
274#461—532no—
275#461—1061no—
276#461—1061no—
277#459C2×D5311yescharacteristic
278#459C2×D5311yescharacteristic
279#447—21no—
280#442—21no—
281#442—21no—
282#436—532no—
283#435—1062no—
284C8×D4—532no—
285#432—531no—
286#432C2×D5311yescharacteristic
287#431—531no—
288C2×C4⋊C8—531no—
289C2×#103—1061no—
290#407—1061no—
291C2×C22⋊C8—1061no—
292#335—532no—
293#334—532no—
294#333—532no—
295#332—532no—
296#332—532no—
297#331—532no—
298C53#1946211yescharacteristic, complemented, Sylow
299C2×#41—21no—
300C22×Q8—1061no—
301C22×D4C22⋊D5311yescharacteristic
302C23×C4—21no—
303C2×SD16—1062no—
304C2×SD16—1062no—
305C2×SD16—2124no—
306C2×D8—21no—
307C2×D8—21no—
308C2×M16—1062no—
309C22×C8—1062no—
310C22×C8—1061no—
311C4⋊Q8—532no—
312C4⋊Q8—1061no—
313C4⋊Q8—1061no—
314C4⋊Q8—1061no—
315C4⋊Q8C22×D5311yescharacteristic
316#127C22×D5311yescharacteristic
317#124—21no—
318#121—21no—
319#121—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4C22⋊D5311yescharacteristic
325C4×D4C22⋊D5311yescharacteristic
326C4×D4C22×D5311yescharacteristic
327C4×D4C22×D5311yescharacteristic
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—1061no—
330C2×C4⋊C4C22⋊D5311yescharacteristic
331C2×C22⋊C4—21no—
332C2×C42C22×D5311yescharacteristic
333#106—21no—
334#106—21no—
335#106—1062no—
336#106—1062no—
337C4⋊C8—532no—
338C4⋊C8—532no—
339C4⋊C8—532no—
340C4⋊C8—1061no—
341C4⋊C8C22×D5311yescharacteristic
342C4⋊C8C22×D5311yescharacteristic
343#103—1062no—
344#103—1062no—
345#102—21no—
346#102—21no—
347#102—21no—
348#102—21no—
349#102—1062no—
350#102—1062no—
351C22⋊C8—21no—
352C22⋊C8—531no—
353C22⋊C8—531no—
354C22⋊C8—1062no—
355C22⋊C8—1062no—
356C22⋊C8—1061no—
357#97—21no—
358#97—1061no—
359C4×C8—532no—
360C24—21no—
361#41—82no—
362C2×Q8—21no—
363C2×Q8—1062no—
364C2×Q8—2122no—
365C2×Q8—2122no—
366C2×Q8—2121no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4C2×C22⋊D5311yescharacteristic
375C2×D4C2×C22⋊D5311yescharacteristic
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—82no—
383C22×C4—2121no—
384C22×C4D4×D5311yescharacteristic
385C22×C4#1171994711yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387SD16—2124no—
388SD16—2124no—
389D8—42no—
390D8—42no—
391M16—4242no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—532no—
395C2×C8—532no—
396C2×C8—1062no—
397C2×C8—1062no—
398C2×C8—4242no—
399C2×C8—2122no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—1062no—
404C4⋊C4—1062no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C42D4×D5311yescharacteristic
416C42#1171994711yescharacteristic
417C42C2×C22⋊D5311yescharacteristic
418C42C23×D5311yescharacteristic
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—81no—
425C23#1313318911yescharacteristic
426Q8—81no—
427Q8—4242no—
428Q8—2122no—
429Q8—2122no—
430D4—21no—
431D4—21no—
432D4—21no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—42no—
437D4—81no—
438D4—82no—
439D4#1313324212yes—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—4242no—
454C2×C4—2122no—
455C2×C4#1313318911yescharacteristic
456C2×C4#1313318911yescharacteristic
457C2×C4#1313318911yescharacteristic
458C2×C4C2×D4×D5311yescharacteristic
459C2×C4C22×C22⋊D5311yescharacteristic
460C2×C4C2×#1171994711yescharacteristic
461C8—41no—
462C8—41no—
463C8—1062no—
464C8—1062no—
465C8—2122no—
466C8—2122no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—84no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22#1681088412yes—
480C22C2×#1313318911yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—81no—
488C4—4242no—
489C4C2×#1313324211yescharacteristic
490C4#1681142111yescharacteristic
491C4#1681143312yes—
492C2—22no—
493C2—42no—
494C2—81no—
495C2—161no—
496C2#2028422911yescharacteristic
497C2#2028433611yescharacteristic
498C2C2×#1681088411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3109C2
socleC22×C53C2×#121
hypercenter#3109D53
hypocenterC53#19462
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20285840C2semidirect
2. C2×#16810828C2semidirect
3. #20283750C2semidirect
4. #20274274C2semidirect
5. #20274216C2semidirect
6. #20274150C2semidirect
7. #20274092C2semidirect
8. #20273501C2semidirect
9. #20273433C2semidirect
10. #20273410C2semidirect
14. #16811019C22semidirect
15. #16810969C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810372C22semidirect
35. #16810370C22semidirect
36. #16810370C22semidirect
37. #16810367C22semidirect
298. C53#19462semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202858402
C2×#168108282
#202837502
#202742742
#202742162
#202741502
#202740922
#202735012
#202734332
#202734102
C53×#31092
#1946253